Bad Math In Good Math

1.0 Introduction and Overview of the Book

Mark C. Chu-Carroll's blog is Good Math, Bad Math. His book is Good Math: A Geek's Guide to the Beauty of Numbers, Logic, and Computation.

A teenager recently asked me about what math he should learn if he wanted to become a computer programmer or game developer. One cannot recommend a textbook (on discrete mathematics?) to answer this, I think. If you do not mind the errors, this popular presentation will do. I like how it presents the building up of all kinds of numbers from set theory. And the order of this presentation seems right, starting with the natural numbers, but then later providing a set theoretic construction in which the Peano axioms were derived. (I suppose Chu-Carroll could also present a complementary explanation of the need for more kinds of numbers by starting out with the problem of finding roots for polynomial equations in which all coefficients are natural numbers. Eventually, you would get to the claim that an nth degree polynomial with coefficients in the complex numbers has n zeros (some possibly repeating) in the complex numbers.)

The book also has an introduction to the theory of computation, with descriptions of Finite State Machines, lambda calculus, and Turing machines. There is an outline of how the universal Turing machine cannot be improved, in terms of what functions can be computed. It doesn't help to add a second or more tapes. Nor does it help to add a two-dimensional tape. The book concludes with a presentation of a function that cannot be computed by a universal Turing machine. The halting problem, as is canonical, is used for an illustration.

2.0 Bad Math Not In Good Math

Besides being interested in popular presentations of mathematics, I was interested in seeing a book developed from blog posts. Chu-Carroll wisely leaves out a large component of his blog, namely the mocking of silly presentations of bad math. I could not do that with this blog. But there is a contrast here. The bad economics I attempt to counter is presented by supposed leaders of the field and heads of supposed good departments. The bad math Chu-Carroll usually writes about is not being to used to make the world a worse place, to obfuscate and confuse the public, to disguise critical aspects of our society. Rather, it is generally presented by people with less influence than Chu-Carroll or academic mathematicians.

2.1 Not a Proof

Anyways, I want to express some sympathy for why some might find some propositions in mathematics hard to accept. I do not want to argue such nonsense as the idea that Cantor's diagonalization argument fails, by conventional mathematical standards; that different size infinities do not exist; or that 0.999... does not equal 1. Anyways, consider the following purported proof of a theorem.

Theorem:

Proof: Define S by the following:

Then a S is:

Subtract a S from S:

Or:

Thus:

The above was what was to be shown.

Corollary: 0.999... = 1

Proof: First note the following:

Some simple manipulations allow one to apply the theorem:

Or:

That is:

2.2 Comments on the Non-Proof and a Valid Proof

I happen to think of the above supposed proof as a heuristic than I know yields the right answer, sort of. A student, when first presented with the above by an authority, say, in high school, might be inclined to accept it. It seems like symbols are being manipulated in conventional ways.

I do not know that I expect a student to notice how various questions are begged above. What does it mean to take an infinite sum? To multiply an infinite sum by a constant? To take the difference between two infinite sums? To define an infinitely repeating decimal number? But suppose one does ask these questions, questions whose answers are presupposed by the proof. And suppose one is vaguely aware of non-standard analysis. Besides how does inequality in the statement of the theorem arise? One might think the wool is being pulled over one's eyes.

How could one prove that 0.999... = 1? First, one might prove the following by mathematical induction:

Then, after defining what it means to take a limit, one could derive the previously given formula for the infinite geometric series as a limit of the finite sum. (Notice that the restriction in the theorem follows from the proof.) Finally, the claim follows, as a corollary, as shown above.

3.0 Errata and Suggestions

I think that this is the most useful part of this post for Chu-Carroll, especially if this book goes through additional printings or editions.

• p. 7, last line: "(n + 1)(n + 2)/n" should be "(n + 1)(n + 2)/2"
• p. 11, 7 lines from bottom: "our model" should be "our axioms".
• p. 19: Associativity not listed in field axioms.
• p. 20: Since the rational numbers are a field, continuity is not part of the axioms defining a field.
• Sections 2.2 and 3.3: Does the exposition of these constructions already presume the existence of integers and real numbers, respectively?
• p. 21: Shouldn't the definition of a cut be (ignoring that this definition already assumes the existence of the real number r) something like (A, B) where:

A = {x | x rational and x ≤ r}

B = {x | x rational and x > r}

• p. 84, footnote: If one is going to note that exclusive or can be defined in terms of other operations, why not note that one of and or or can be defined in terms of the other and not? Same comment applies to if ... then.
• p. 85, last 2 lines: the line break is confusing.
• p. 95, proof by contradiction of the law of the excluded middle: Is this circular reasoning? Maybe thinking of the proof as being in a meta-language saves this, but maybe this is not the best example.
• p. 97, step 1: Unmatched left parenthesis.
• p. 106: Definition of parent is not provided, but is referenced in the text.
• p. 114, base case: Maybe this should be "partition([], [], [], []).
• p. 130: In definitions of union, intersection, and Cartesian product, logical equivalence is misprinted as some weird character. This misprinting seems to be the case throughout the book (e.g., see pp. 140, 141, and 157).
• p. 133 equation: Right arrow misprinted as ">>".
• Chapter 17: Has anybody proved ZFC consistent? I thought it was the merely the case that nobody has found an inconsistency or can see how one would come about.
• p. 148: Might mention that the order being considered in the well-ordering principle is NOT necessarily the usual, intuitive order.
• p. 148: Drop "larger" in the sentence ending as "...there's a single, unique value that is the smallest positive real number larger!"
• p. 163" "powerset" should be "power set".
• p. 164, line 6: "our choice on the continuum as an axiom" is awkward. How about, "our choice about the continuum hypothesis as an axiom"?
• p. 168, Table 3: g + d = e should be g + d = g.
• p. 171-172: Maybe list mirror symmetry or write, "in addition to mirror symmetry".
• Part VI: Can we have something on the Chomsky hierarchy?
• p. 185; p. 186, Figure 15; p. 193): Labeling state A as a final state is inconsistent with the wording on p. 185, but not the wording on p. 193. On p. 185, write "...that consist of any string containing at least one a, followed by any number of bs."
• p. 190: Would not D_a(ab*) be b*, not ab*?
• p. 223: "second currying example" should be "currying example". No previous example has been presented.
• p. 225, towards bottom of page: I do not understand why α does not appear in formal definition of β.
• p. 229: Suggestion: Refer back to recursion in Section 14.2 or to chapter 18.
• p. 244, 5 lines from bottom: Probably γ should not be used here, since γ was just defined to represent Strings, not a generic type. Same comment goes for α.
• p. 245, last bullet: It seems here δ is being used for the boolean type. On the previous page, β was promised to be used for booleans, as in the first step of the example on the bottom of p. 247.
• p. 249 (Not an error): The reader is supposed to understand what "Intuitionistic logic" means, with no more background than that?
• p. 257: Are the last line of the second paragraph and the last line of the page consistent in syntax?
• Can we have an index?

What Is A "Special Interest"?

I do not want to compare and contrast analytically precise definitions that answer the question in the post title. (Socrates, as reported by Plato, always asked for a definition after being given examples.) Instead, I give two lists, where I trust the reader to see family resemblances among the items on each list:

• Ethnic groups like African-Americans; women; the poor; organized labor; and lesbians, gays, bisexuals, and transgenders.
• Corporations, especially those operating in specific industries (e.g., big oil); Corporate Executive Officers; and owners of small businesses.

I suggest that the policies and culture of a country would be quite different, when the dominant understanding of the phrase, "special interests" was consistent with one or another list.

I think somewhere or other Noam Chomsky has asserted that the second understanding reflects the true meaning or the term, or at least a meaning consistent with what the Founding Fathers of the United States wrote. This quote does not have the look back to classical liberals:

"...these questions have been asked for a long time in polls, a little differently worded so you get some different numbers, but for a long time about half the population was saying, when asked a bunch of open questions - like, Who do you think the government is run for? would say something like that: the few, the special interests, not the people. Now it's 82%, which is unprecedented. It means that 82% of the population don't even think we have a political system, not a small number.

What do they mean by special interests? Here you've got to start looking a little more closely. Chances are, judging by other polls and other sources of information, that if people are asked, Who are the special interests? they will probably say, welfare mothers, government bureaucrats, elitists professionals, liberals who run the media, unions. These things would be listed. How many would say, Fortune 500, I don't know. Probably not too many. We have a fantastic propaganda system in this country. There's been nothing like it in history. It's the whole public relations industry and the entertainment industry. The media, which everybody talks about, including me, are a small part of it. I talk about mostly that sector of the media that goes to a small part of the population, the educated sector. But if you look at the whole system, it's just vast. And it is dedicated to certain principles. It wants to destroy democracy. That's its main goal. That means destroy every form of organization and association that might lead to democracy. So you have to demonize unions. And you have to isolate people and atomize them and separate them and make them hate and fear one another and create illusions about where power is. A major goal of this whole doctrinal system for fifty years has been to create the mood of what is now called anti-politics." -- Noam Chomsky, Class Warfare: Interviews with David Barsamian Common Courage Press (1966): p. 138.

But there is another literature, a post modern literature, that also looks at how people come to associate examples with words. People generally do not think logically, following the rules of predicate calculus. One trying to understand culture should realize this. One might talk about the The politics of the signifier. How does one or another definition, or set of examples, become hegemonic? (For what it is worth, I think Slavoj Zizek is a very intelligent, very well-read, self-aware clown.)

Income Inequality In OECD Countries

I recently took another look at data, available from the Organization for Economic Co-operation and Development (OECD), on income inequality. The Gini coefficient is available on countries in the database, under measures of Social Protection and Well-being. Under that menu, expand the sub menu for Income distribution and poverty, and select inequality. You can see the Gini coefficient (at disposable income, post taxes and transfers) displayed, by country, for various years. Table 1 shows the most recent numbers, sorted from countries with the most equal distribution to the least equal. For one way of thinking about it, the United States is not number 1, since the US is exceeded by Turkey, Mexico, and Chile.

Table 1: Gini Coefficient

Country	Gini Coefficient (Non Provisional)	Year
Slovenia	0.245	2011
Norway	0.250	2011
Iceland	0.251	2011
Denmark	0.253	2011
Czech Republic	0.256	2011
Finland	0.261	2012
Slovak Republic	0.261	2011
Belgium	0.264	2010
Sweden	0.273	2011
Luxembourg	0.276	2011
Netherlands	0.278	2012
Austria	0.282	2011
Switzerland	0.289	2011
Hungary	0.290	2012
Germany	0.293	2011
Poland	0.304	2011
Korea	0.307	2012
France	0.309	2011
Ireland	0.312	2009
Canada	0.316	2011
Italy	0.321	2011
Estonia	0.323	2011
New Zealand	0.323	2011
Australia	0.324	2012
Greece	0.335	2011
Japan	0.336	2009
United Kingdom	0.341	2010
Portugal	0.341	2011
Spain	0.344	2011
Israel	0.377	2011
United States	0.389	2012
Turkey	0.412	2011
Mexico	0.482	2012
Chile	0.503	2011

The Gini coefficient is a measure of inequality, with a higher Gini coefficient denoting a more unequal distribution of income. It is defined as follows: sort the population in order of increasing income. Plot the percentage of income received by those poorer than each value of income against the percentage of the population with less than that value of income. This is the Lorenz curve, and it will fall below a line with a slope of 45 degrees going through the origin. The Gini coefficient is the ratio of the area between the 45 degree line and the Lorenz curve to the area under the 45 degree line. A Gini coefficient of zero indicates perfect equality, while a Gini coefficient of unity arises when one person receives all income and everybody else gets nothing. Consequently, the Gini coefficient lies between zero and one.

A Cynical Take By Greece's Finance Minister On Mainstream Economists

I have found Yanis Varoufakis' 2014 book, Economic Indeterminacy: A personnel encounter with economists' peculiar nemesis a bit too abstract for my tastes. I am not sure that game theory counts as a subset of neoclassical economics, although I can see how some game theory meets Varoufakis' definition. One might see how a lot of game theory illustrates the idea that economists, collectively, exhibit weakness of will. That is, a lot of game theory can be used to develop models with multiple equilibria and of nondeterministic outcomes. One might expect economists to shy away from these conclusions.

I find it hard to accept Varoufakis's argument that in games, one might want to deliberately be irrational. I wondered if that was so, wouldn't an opponent see this? And, thus, would not this irrational behavior therefore be rational at a meta-level? Varoufakis' argument is structured to address this objection.

But my point in this post is to quote from the preface:

"...my project's failure was predetermined, at least in the sense that it was never going to cause a shift in the attitudes and demeanour of a profession which operates like a priesthood, dedicated solely to preservation of its dogmas... as well as to the recapitulation of its authority within the universities, the financial sector and the government. Indeed, at no point did I harbour any significant hope that this priesthood would take kindly to the demons of doubt and indeterminacy which my work was bound to give rise to. But it did not matter, at least not at a personal level. My intimate familiarity with the neoclassical models was sufficient to keep me on the roster of neoclassical economics departments, where a capacity to teach these models, and produce academic papers based on them is all that matters.

Looking back at these long years of tampering with, and delving into, the complex models of the neoclassical tradition, I cannot but question my decision to keep pushing, Sisyphus-like, the theoretical rock up the neoclassical hill. Why did I stick to this task, when I knew it would end up in failure? In retrospect, there were two reasons, neither of which was predicated upon any hope of influencing a profession utterly uninterested in the truth status of its models. First, I deeply enjoyed toying with these models as an end-in-itself, just as a clockmaker enjoys taking apart and then re-assembling some old clock for the hell of it. Secondly, and more importantly, I felt it necessary to make it hard for my colleagues to pretend to themselves that the models they were being forced to with, by a particularly authoritarian profession, were logically coherent. Bringing them, even fleetingly, to the point when they had to confess to their models' internal contradictions was, I felt, a victory of sorts; the equivalent of a lone sniper behind enemy lines making life difficult for an army of cocupation." -- Yannis Varoufakis (2004: p. xxiv.)

Varoufakis has some other books that sound interesting and more popular. I think his book; The Global Minotaur: America, Europe and the Future of the Global Economy; might be especially topical at the moment.

Update: Steve Keen provides a link to one exposition of Varoufakis' argument that, in game theory, agents can and will deliberately choose irrational behavior.

Approximating a Continuous Time Markov Process

Figure 1: Rate of Transitions Between States in a Three-State Markov Chain

1.0 Introduction

This post, about Markov processes, does not have much to do with economics. I here define how to approximate a continuous time Markov chain with a discrete time Markov chain. This mathematics is useful for one way of implementing computer simulations involving Markov chains. That is, I want to consider how to start with a continuous time model and synthesize a realization with a small, constant time step.

2.0 Continuous Time Markov Chains

Consider a stochastic process that, at any non-negative time t is in one of N states. Assume this process satisfies the Markov process: the future history of the process after time t depends only on the state of the process at time t, independently of how the process arrived at that state. I consider here only processes with stationary probability distributions for state transitions and for times between transitions. A continuous time Markov chain is specified by a state transition matrix. In this section, I define such a matrix as well as specifying two additional assumptions.

Formally, let P_{i, J} denote the conditional probability that the next transition will be into state j, given that the process is in state i at time zero. (As seen below, in the notation adopted here it matters that these conditional probabilities are not a function of time.) Assume that for each state, the next transition when the process is in that state is into a different state:

P_{i, i} = 0; i = 0, 1, ..., N - 1

Further, assume that for each state, the time to the next transition is from an exponential distribution with the rate of occurrence of state transitions dependent only on the initial state:

F_{i, j}(t) = 1 - e^{- λ_i t}; i, j = 0, 1, ..., N - 1;

where F_{i, j}(t) is the conditional probability that the next transition will be before time t, given that the chain is in state i at time zero and that the next transition will be into state j. In other words, F_{i, j}(t) is the Cumulative Distribution Function (CDF) for the specified random variable. Under the above definitions, the stochastic process is a continuous time, finite state Markov chain.

Let P_{i, j}(t) be the conditional probability that the chain is in state j at time t, given that the chain is in state i at time zero. These conditional probabilities satisfy Kolmogorov's forward equation:

,

where the transition rate matrix Q is defined to be:

The elements in each row of the transition rate matrix sum to zero. Kolmogorov's forward equation can be expressed in scalar form:

The above equation applies to continuous time Markov chains with a countably infinite number of states only under certain special conditions.

Steady state probabilities of this Markov chain satisfy:

p Q = 0,

where p is a row vector in which each element is the steady-state probability that the chain is in the corresponding state.

3.0 Discrete Time Approximation

A discrete time Markov chain is specified by a state transition matrix A, where a_{i, j} is the probability that the chain will transition in a time step from state i to state j, given that the chain is in state i at the start of the time step. Steady state probabilities for a discrete time Markov chain satisfy:

p A = p

The above equation compares and contrasts with how steady state probabilities relate to the transition rate matrix in a continuous time Markov chain.

Let the time step h be small enough that the probability of the continuous time Markov chain undergoing two or more transitions in a single time step is negligible. In other words, the following probability, calculated from a Poisson distribution, is close to unity for all states i:

P(0 or 1 transitions in time h | Chain in state i at time 0) =

(1 + λ_i h) e^{- λ_i h}

The probability that the chain remains in a given state for a time step is the probability that no transitions occur during that time step, given the state of the chain at the start of the time step. This probability is also found from a Poisson distribution:

a_{i, i} = e^{- λ_i h} = e^{- q_{i, i} h}; i = 0, 1, ..., N - 1

The probability that the chain transitions to state j, given the chain is in state i at the start of the time step, is the product of:

• The probability that a transition occurs during that time step, and
• The conditional probability that the next transition will be into state j, given the chain is in state i at the start of the time step.

The following equation specifies this probability:

a_{i, j} = (1 - a_{i, i})P_{i, j} = (1 - a_{i, i}) q_{i, j}/(- q_{i, i}); i ≠ j

These equations allow one to write a computer program to synthesize a realization from a finite state Markov chain, given the parameters of a continuous time, finite state Markov chain. Such a program will be based on a discrete time approximation.

4.0 An Example

Consider a three-state, continuous time Markov chain. Figure 1 shows the rate of transitions between the various states. The transition rate matrix is:

To discretize time, choose a small time step h such that, for all states i, the following probabilities are approximately unity:

P(0 or 1 transitions in time h | Chain in state 0 at time 0) =

[1 + (λ_{0, 1} + λ_{0, 2})h] e^{-(λ_{0, 1} + λ_{0, 2})h}

P(0 or 1 transitions in time h | Chain in state 1 at time 0) =

[1 + (λ_{1, 0} + λ_{1, 2})h] e^{-(λ_{1, 0} + λ_{1, 2})h}

P(0 or 1 transitions in time h | Chain in state 2 at time 0) =

[1 + (λ_{2, 0} + λ_{2, 1})h] e^{-(λ_{2, 0} + λ_{2, 1})h}

The state transition matrix A for the discrete-time Markov chain is:

I have not tested the above with concrete values for a continuous time Markov chain.

Reference

• S. M. Ross (1970). Applied Probability Models with Optimization Applications. San Francisco: Holden-Day

Laughing At Neoclassical Economists, Elsewhere

• Matthew Yglesias lists "Nine Things Only Neoclassical Economists Will Understand". Strangely, his twitter announcement of this article is about a tenth.
• Noah Smith purports to explain each thing in only a couple of sentences. Stranegly, only for the Modiliani-Miller theorem does he note, "Obviously this doesn't work in the real world".
• Tyler Cowen attempts to clarify the Heckscher-Ohlin theorem, but fails to note that "capital" cannot be a factor of production in the Heckscher-Ohlin-Samuelson model. (He does note Leontief's empirical demonstration that the theory fails.)

Because Something Is Happening Here/But You Don’t Know What It Is/Do You, Mister Jones?

Strangely, some prominent, somewhat liberal, economics bloggers have decided simultaneously to complain about (unnamed) left-leaning heterodox economists:

• Noah Smith, at Bloomberg View and at his blog.
• Simon Wren-Lewis, at his blog.
• Paul Krugman, at his New York Times blog.

All three, incorrectly in my view, think the heterodox economists who they object to are only arguing politics. As far as I know, many, including me, do not take issue with Krugman's short-term policy views. Smith, in his trollish approach, raises a side comment about Austrian economists and the Mont Pelerin society. (I will state the proper label for Friedrich Hayek and Ludwig Von Mises is "economist", not "quasi-economist", as Smith would have it. But I've seen for some time that I am more well-informed on Austrian economics than Smith is.)

I think more pertinent issues center around modeling approaches, the image the profession projects in the public sphere, and the sociology of the profession. How is it than so many rightists have been able to push the view that their politics is good economics, while simultaneously insisting that economics is a positive science? The involvement of economists with neoliberal politics is not confined to some fringe. Consider, for example, the Chicago school, the lack of a strong ethics policy in the American Economic Association, the Washington consensus, and even Paul Samuelson's 1960s research that led to to Efficient Market Hypothesis.

There is probably also a personnel element here. Non-mainstream, heterodox economists would like more acknowledgement by mainstream economists. If your knowledge of heterodox economics is confined to what you can get off the Internet, aside from what professional literature is now available there, you might not know what you are talking about when you talk about heterodox economics. (And this includes the Austrian school.) Furthermore, when you develop parallel ideas, or draw on heterodox economics, you should acknowledge it. In the linked post above, Krugman makes the point that "a country that borrows in its own currency" cannot easily become like Greece, under attack from "bond vigilantes", without saying anything about endogenous money or the economists at the University of Missouri Kansas City. (I could also say something about the research for which Krugman won the "Nobel Prize".) If you know where to look, you can find Joseph Stiglitz acknowledging that he learned a lot from such Cambridge economists as Nicky Kaldor and Joan Robinson.

Maybe economics would be a better place if the center of gravity in economics in the United States were arguments between mainstream economists and, say, economists at the New School and the University of Massachusetts at Amherst. If the profession were to move in this direction, young doctorates would need to be socialized to not dismiss viewpoints because of the rankings of the universities and the journals in which they were advanced. Methodology would continue to need to be broadened to include more than mathematical models of optimizing agents.

Update: Reactions from Chris Dillow, Peter Dorman, and Alex Marsh.

Greg Mankiw, Fool Or Knave?

Greg Mankiw seems determined to continually attempt to bring his supposed profession into disrepute. Last week, at the annual meeting of American economists (the Allied Social Science Associations), Greg Mankiw chaired a session on Thomas Piketty's Capital in the 21st Century. In his draft of his prepared remarks, Mankiw writes:

"Equation (3) says that capital earns its marginal product." -- Greg Mankiw, "Yes, r > g. So what?" (24 November 2014).

Because of price Wicksell effects, the marginal product of finance capital is generally unequal, in equilibrium, to the rate of profits. Even Champernowne's chain index, which abstracts from price Wicksell effects, cannot generally be used to defend the equality in aggregate models of the rate of profits and the marginal product of capital. Economic theory imposes no restriction on the direction of price and real Wicksell effects, and the chain index is not well-defined in the presence of positive Wicksell effects. Neoclassical theory claims, at best, that the price of each capital good is equal, in equilibrium, to its marginal product. But marginal productivity is not a theory of the functional income distribution, since every point on the wage-rate of profits frontier is compatible with all valid marginal productivity conditions. Even if the returns to capital could be explained by marginal productivity, this would not justify any particular size of the tolls that capitalists are able to impose. A conceptual distinction can and should be made between the cost of capital goods and the returns to capitalists.

As far as I am concerned, the above is just good economics, agreed to by all non-ignorant economists, neoclassical or otherwise. But the confusion and general muddleheadness promoted by such as Mankiw, seems to serve a functional purpose in the sublunary world.

On "Privatized Keynesianism"

I have been reading Colin Crouch's The Strange Non-Death of Neoliberalism¹. A major theme is that an ideological divide between more reliance on markets and on government misses issues raised by the existence of large - including multinational - corporations. The neoliberal assault on government has been increasing the strength of corporations, not competitive markets. Furthermore, corporations have been taking on the role of government. Crouch mentions, for example, the "seconding" of corporate executives to various ministries; the likelihood that internal policies of a Multi-National Corporation on, say, child labor may be more restrictive than laws in many third world countries; and the role of corporations in setting international standards, where organizations with nation-states may be weak.

But my point in this post is to note Crouch's introduction(?) of a new technical term, Privatized Keynesianism. A contrast between the post-World War II golden age and the later neoliberal era² is needed to make sense of this term. After the war, in the United States - and, I gather, in other advanced industrial capitalist economies - wages rose with average productivity. Furthermore, governments, under a somewhat Keynesian ideology, saw it as their responsibility to maintain aggregate demand. These conventions came undone in the 1970s. Productivity increased (at a slower pace), but wages failed to keep up, and governments came to emphasize fighting inflation, not unemployment.

Increased inequality, however, did not eliminate the need to manage aggregate demand. Neither consumer spending from wages nor an abdication from fiscal polity by government could fill this lacuna. This period saw the increased availability of debt, the creation of secondary markets for the trading of bets on bets on bundles of debts (derivatives), and the capture of credit rating agencies by sellers of debts. This institutional structure led to the collective, but private, macroeconomic regulation of aggregate demand³. This institutional structure is what Crouch calls privatized Keynesianism⁴. The irresponsibility of banks, in some sense, produced a (temporary, unsustainable) positive externality.

Footnotes

1. I might as well note two mistakes I found irritating. Somewhere in one of the early chapters, Crouch, who I gather is British, refers to Eugene McCarthy when he means Joe McCarthy. I also thought that Crouch's account of the role of Fanny Mae and Freddy Mac in subprime mortages reflected too much credence for right-wing liars.
2. I date the start of the neoliberal era with Nixon ending the fixed exchange rate between the United States dollar and gold, a major element of the Bretton Woods system.
3. Is this a non-microfounded, functionalist account?
4. From this perspective, the accumulation of private debt was a symptom, not the ultimate cause of the recent Global Financial Crisis, a cause that has yet to be addressed. These ideas seem to me to be close to Thomas Palley's Structural Keynesianism. Has anybody read James K. Galbraith's The End of Normal: The Great Crisis and the Future of Growth?

Slaves Identifying With Their Masters

Marx's attempt to describe how capitalism creates objective illusions, so to speak, is one aspect of Capital that I like. In this comment on a long-ago Crooked Timber post, "Ted" draws an analogy to J. S. Mill's Subjection of Women, which I have never read. Apparently, Mill explains how women can come to identify with their oppressors.

I happen to currently be reading the autobiography of local Rochester hero, Frederick Douglass. This passage identifies a curious phenomenon:

"Moreover, slaves are like other people, and imbibe prejudices quite common to others. They think their own better than that of others. Many, under the influence of this prejudice, think their own masters are better than the masters of other slaves; and this, too, in some cases, when the very reverse is true. Indeed, it is not uncommon for slaves even to fall out and quarrel among themselves about the relative goodness of their masters, each contending for the superior goodness of his own over that of the others. At the very same time, they mutually execrate their masters when viewed separately. It was so on our plantation. When Colonel Lloyd's slaves met the slaves of Jacob Jepson, they seldom parted without a quarrel about their masters; Colonel Lloyd's slaves contending that he was the richest, and Mr. Jepson's slaves that he was the smartest, and most of a man. Colonel Lloyd's slaves would boast his ability to buy and sell Jacob Jepson. Mr. Jepson's slaves would boast his ability to whip Colonel Lloyd. These quarrels would almost always end in a fight between the parties, and those that whipped were supposed to have gained the point at issue. They seemed to think that the greatness of their masters was transferable to themselves. It was considered as being bad enough to be a slave; but to be a poor man's slave was deemed a disgrace indeed." -- Frederick Douglas, Narrative of the Life of Frederick Douglas

Maybe some day I'll read Hegel's Phenomenology of Spirit - it is on my shelf - to learn some ideas about the master-slave dialetic. Mayhaps the above is analogous to the opinions of many wage-slaves. There seem to be many ways to be unfree, and many ways to deny this.

First Formulation of Folk Theorem and Indeterminacy in Game Theory

Initial and Chaotic Learning in Rock-Paper-Scissors

Consider a game, as games are defined in game theory. And consider some strategy for some player in some game. The folk theorem states, roughly, that any strategy can be justified as a solution for a game by considering an infinitely repeated game. (An amusing corollary might be stated as saying that competition is the same as monopoly, if you do the math right.) The following seems to me to state the folk theorem (abstracting from the distinction between Nash equilibria and Von Neumann and Morgenstern's solution concept):

"21.2.3. If our theory were applied as a statistical analysis of a long series of plays of the same game - and not as the analysis of one isolated play - an alternative interpretation would suggest itself. We should then view agreements and all forms of cooperation as establishing themselves by repetition in such a long series of plays.

It would not be impossible to derive a mechanism of enforcement from the player's desire to maintain his record and to be able to rely on the on the record of his partner. However, we prefer to view our theory as applying to an individual play. But these considerations, nevertheless, possess a certain signiificance in a virtual sense. The situation is similar to the one we encountered in the analysis of the (mixed) strategies of a zero-sum two-person game. The reader should apply the discussions of 17.3 mutatis mutandis to the present situation." -- John Von Neumann and Oscar Morgenstern (1953) p. 254.

I have heard it claimed that economic theory has developed such that any moderately informed graduate student can now provide you with a model that yields any conclusion that you like. The folk theorem, as I understand it, is not even the most threatening finding for the ability of game theory to yield determinate conclusions.

Consider an iterated game before an equilibrium, under some definition or another, has been achieved. The players are trying to learn each others' strategies. Even a simple game, such as Rock-Scissors-Paper, can yield chaotic dynamics (Sato, Akiyama, and Farmer 2002; Galla and Farmer 2013). An equilibrium might never be established, for it is worthwhile for some players to deliberately choose "irrational" moves so as to ensure that other players do not achieve equilibrium, instead of a result that benefits the supposedly irrational player (Foster and Young 2012). (I hope I found this reference from reading Yanis Varoufakis, who, in one paper in one of his books, makes this point with the centipede game.) Apparently, this irrationality does not disappear by moving towards a more meta-theoretic level. And one player, who understands the evolutionary behavior of the other player in a Prisoner's Dilemma, can manipulate the other player to result in a asymmetric result - that is, a case where the non-evolutionary player extorts the player following a mindless evolutionary strategy (Press and Dyson 2012, Stewart and Plotkin 2012).

References

• Dean P. Foster and H. Peyton Young (2001). On the impossibility of predicting the behavior of rational agents. Proceedings of the National Academy of Sciences. V. 98: pp. 12848-12853.
• Tobias Galla and J. Doyne Farmer (2013). Complex dynamics in learning complicated games. Proceedings of the National Academy of Sciences, V. 110: pp. 1232-1236.
• Tiago P. Peixoto and Stefan Bornholdt (2013). No need for conspiracy: Self-organized cartel formation in a modified trust game.
• William H. Press and Freeman J. Dyson (2012). Iterated Prisoner's Dilemma contains strategies that dominate any evolutionary opponent. Proceedings of the National Academy of Sciences.
• Yuzuru Sato, Eizo Akiyama, and J. Doyne Farmer (2002). Chaos in learning a simple two-person game. Proceedings of the National Academy of Sciences. V. 99: pp. 4748-4751.
• Alexander J. Stewart and Joshua B. Plotkin (2012). From extortion to generosity, evolution in the iterated Prisoner's Dilemma. Proceedings of the National Academy of Sciences.
• Yanis Varoufakis (2005). Rational Rules of Thumb in Finite Dynamic Games: N-person Backward Induction with Inconsistenly Aligned Beliefs and Full Rationality. American Journal of Applied Sciences.
• Yanis Varoufakis (2013). Economic Indeterminacy: A personal encounter with the economists' peculiar nemesis.
• John Von Neumann and Oscar Morgenstern (1953). Theory of Games and Economic Behavior, third edition. Princeton University Press.

Noah Smith Befuddling Bloomberg Readers

1.0 Introduction

Noah Smith seems to be trying to become a professional columnist and blogger, however his day job works out. I do not know if the same opportunity still exists, as it apparently did when, for example, Duncan Black, Kevin Drum, Ezra Klein, Josh Marshall, Heather Parton, and Matthew Yglesias were starting out. I do not want to spend much time taking down Smith, but I wish so many of his columns did not provide anecdotal evidence that the job of mainstream economists is to sow confusion into the public sphere. Maybe I should try to resolve not to read him.

2.0 Confusion on Marginal Productivity

Consider this Bloomberg column, "You want a bigger paycheck? Convince me." Smith's column contains the, I guess, still obligatory confused red-baiting:

"No economic model says that people get paid based on average productivity. If they did, there would be no income left over for capital -- no profits, rents or interest. We’d be living in a sort of a Marxist world, where labor is the only thing with any value." -- Noah Smith

I do not see what that comment has to do with Marxism. (Consider the Critique of the Gotha Program.) Anyways, this comment immediately follows Smith's graphical and empirical demonstration that real wages rose with increases in productivity in the United States during the post war golden age. Was the United States in the 1950s and 1960s a "sort of Marxist" society? Certainly economic models of growth and distribution exist for thinking about the relationship between wages and average productivity in the golden age, and the breakdown of this relationship in the subsequent neoliberal era.

Smith apparently thinks that the theory of marginal productivity is a theory of the distribution of income. He is, of course, quite mistaken. Even worse, Smith goes on to use the discredited Solovian growth model, with an aggregate Cobb-Douglas production function, to explain how economists supposedly explain (changes in) the shares of "capital" and labor in national income.

Is it progress that Smith does not bring up skills-biased technical change, a nonsensical theory often used to propagandize for increased inequality in the distribution of wages? Maybe not, for Smith's purpose seems to be to propagandize for increased inequality in the functional income distribution between "capital" and wages. And so he brings up an equally nonsensical theory about the "rise of robots".

3.0 Inadequate Understanding on Women in Economics

Even when I don't necessarily disagree with Smith, I often find his columns insufficiently informed. Here he writes about career prospects in economics for women. I thank Smith for bringing this paper by Ceci, Ginther, Kahn, and Williams to my attention. But it takes Claudia Sahm, in a response to this column, to bring up the Committee on the Status of Women in the Economics Profession (CSWEP). And, as far as I am aware, nobody previously commenting on Smith has mentioned the International Association for Feminist Economics (IAFFE) and their journal, Feminist Economics. If you want to argue that homo economicus is gendered, I suggest browsing back issues of that journal.

Humans And Other Animals

Figure 1: Chapuchin Monkeys, Our Cousins

What do we think about generalizations, validated partly with experiments with non-human animals, for economics?

Nicholas Georgescu-Roegen is an economist widely admired by heterodox economists. He quit the American Economic Association in response to their flagship publication, the American Economic Review, publishing articles on, if I recall correctly, pigeons. Researchers were trying to demonstrate that properly trained pigeons had downward-sloping demand curves. I gather they wanted to show income effects and substitution effects, as well, with these laboratory experiments.

On the other hand, are we not supportive of behavioral economists undermining utility theory? I am thinking of controlled experiments that demonstrate people do not conform to the axioms of preference theory. And some of these experiments, as illustrated in the YouTube video linked above, extend beyond humans.

I have a suggestion to resolve such a tension. One might want to treat investigations of humans as a naturalistic enterprise. If so, one would not want to impose an a priori boundary on the different constituents of minds. Whether some species of animals has some sense of self, expectations of the future, primitive languages, or what not should be found by empirical investigation. On the other hand, activities that depend on the existence of social institutions cannot be expected to be found in animals not embedded in any society. And demand curves, if they were to exist, would only arise in specific market institutions.

Reference

• Philip Mirowski (1994). The realms of the Natural, in Natural Images in Economic Thought (ed. by P. Mirowski), Cambridge University Press.

Income Distribution And A Simple Labor Theory Of Value

I have a new paper available on the Social Science Research Network:

Title: Income Distribution And A Simple Labor Theory Of Value: Empirical Results From Comprehensive International Data

Abstract: This paper presents the results of an empirical exploration, with data from countries worldwide, of Sraffian, Marxian, and classical political economy. Income distribution, as associated with systems of prices of production, fails to describe many economies. Economies in most countries or regions lie near their wage-rate of profits frontier, when the frontier is drawn with a numeraire in proportions of observed final demands. Labor values predict market prices better than prices of production do. Labor values also predict market prices better than they predict prices of production. In short, a simple labor theory of value is a surprisingly accurate price theory for economies around the world.

For Conflating Neoliberalism And Neoclassical Economics

Neoliberalism is a political project to remake the world into an unrealizable utopia. Neoclassical economics is a supposedly scientific effort to explain the world by its deviations from an unrealizable utopia. And they are both about how the world deviates from that utopia. This post is about this resemblance, not the differences, between neoliberalism and neoclassical economics.

This utopia consists of a society organized around markets¹. These markets require government to define property rights and enforce contract law. But, in the utopia, they are not to be embedded in a broader institutional setting that prevents their supposedly free adjustment. Examples of government-imposed inference with such self-regulation include minimum wages, rent control, laws against price-gouging, usury laws, subsidies for farmers to limit the size of harvests so as to maintain their income, payments to the able-bodied unemployed², and so on. Polanyi's claim is that such so-called interventions are bound to arise. The ideal which those enacting such laws were reacting against is unachievable, anyways. In the ideal, land, labor, and capital are treated as if they are only commodities. But land is the natural setting in which the economy takes place, and labor and capital involve social relations that cannot be reduced only to market relationships.

Both neoliberals and neoclassical economists often recognize their utopia must be constructed³, that it, will not emerge naturally, in some sense. The solution for problems with markets is said to be to construct more markets. I think about the tragedy of the commons, the theory of externalities^{4, 5}, and the emphasis in neoclassical welfare theory on Pareto optimality. A paradigmatic policy recommendation, for both neoliberals and neoclassical economics, is the establishment of markets for pollution permits.

Footnotes

1. I have been reading Block and Somers (2014), and I read Polanyi (1944) more than a decade ago.
2. Block and Somers approvingly cite revisionist history from Mark Blaug in the 1960s that challenged centuries-long interpretations of English Poor Laws, especially the Speedhamland system. I know Blaug through his (multi-edition) history of economics and his misrepresentations of Sraffians and the Cambridge Capital Controversy. So I was glad to see a cite where he seems to be correct.
3. This emphasis on the need for government to construct markets, to my mind, is a distinctive difference between classical liberals and sophisticated neoliberals.
4. Some mainstream economists defend themselves from critics by asserting that the critics attack a strawperson. Economists do not believe, they say, that markets are perfect. And they'll ask why are the critics not aware of the frequent teaching about externalities. This objection seems to me to be beside the point if neoclassical economists react, as many do, the existence of an externality by calling for policy for internalizing the externality (or, at least, imitating the result of such policies).
5. If one accepted neoclassical economics as a positive science, how could one call for any policy conclusion without an explicit statement of normative values at some low level of abstration?

References

• Fred Block and Margaret R. Somers (2014). The Power of Market Fundamentalism: Karl Polanyi's CritiqueHarvard University Press.
• Karl Polanyi (1944). The Great Transformation: The Political and Economic Origins of Our Time.

Fred Lee

Barkley Rosser, David Ruccio, and Matias Vernengo have obituaries. I find I had not known much about Lee's life.

I have been influenced by Lee's work on markup pricing (also known as full-cost pricing), the history of heterodox economics, and the suppression of heterodox economics by the mainstream through bullying and bureaucratic measures. I think highly of Lee's 2004 paper (written in collaboration with Steve Keen), "The Incoherent Emperor: A Heterodox Critique of Neoclassical Microeconomic Theory". I can only find one blog post of mine referencing this paper. Lee promoted pluralism in economics.

Marginal Productivity Theory of Distribution: Acknowledged Blatherskite

I was surprised at how many reviews of Thomas Piketty's Capital in the 21st Century draw on the Cambridge Capital Controversy to argue that Piketty's theoretical framework is grossly inadequate.

• James K. Galbraith's Spring 2014 review in Dissent.
• Dean Baker last May.
• October 2014 special issue of the Real-World Economics Review.
• Tony Aspromourgos's Thomas Piketty, the future of capitalism and the theory of distribution: a review essay (October 2014) (H/T: David Fields).
• Javier Lopez Bernardo, Felix Lopez Martinez, and Engelbert Stockhammer's A Post-Keynesian Response to Piketty's 'Fundamental Contradiction of Capitalism' (October 2014) (H/T Lars P. Syll). This response brings up, in addition to the CCC, the Post Keynesian theory of distribution.
• John Bellamy Foster and Michael D. Yates' review
in Monthly Review

I like this Aspromourgos quote:

However classical the questions Piketty addresses, when he turns to explain the determination of r he has recourse to the conventional, post-classical marginal productivity theory of distribution: diminishing marginal capital productivity is 'natural' and 'obvious' (212–16). (He is much less willing to have recourse to time preference: 358–61; cf. 399–400.) The logical critique of capital aggregates – applied either at the macro or micro level – as supposed independent explanatory variables in the theory of profit rates, first coherently stated by Piero Sraffa (1960, pp. 81–7; see also Kurz and Salvadori 1995, pp. 427–67), is nowhere acknowledged or addressed. That such a relatively well-read economist as Piketty can so unhesitatingly apply this bankrupt approach, is testament to how completely a valid body of critical theoretical analysis can be submerged and forgotten in social science (a phenomenon for the sociologists of knowledge to contemplate). This is so, notwithstanding that Piketty offers a brief interpretation of the 'Cambridge' capital debates, making them turn upon the issues of whether there is substitutability in production (and associated flexibility of capital-output ratios), and whether or not 'growth is always perfectly balanced [i.e., full-employment growth]' (230–32). In fact, the participants on both sides of those debates were concerned with production systems in which substitution and capital-output variability occurred; and continuous full-employment growth was not entailed by recourse to orthodox, marginalist production functions, a point perfectly understood by the participants on both sides. -- Tony Aspromourgos

Update (27 October 2014): Added the Bernardo, Martinez, and Stockhammer reference.

Update (1 December 2014): Added the Foster and Yates reference.

r > g In A Steady State

1.0 Introduction

This post presents a model of distribution that Luigi Pasinetti developed. It is one of a family of models. Other important models in this family were developed by Richard Kahn, Nicholas Kaldor, and Joan Robinson. These models have been extended in various ways and presented in textbooks. One can see this family as having extended work by Roy Harrod, and as being related to the work of Michal Kalecki and even of Karl Marx.

2.0 The Model

2.1 Definitions

Consider a simple closed economy with no government. All income is paid out in the form of either wages or profits:

Y = W + P,

where W is total wages, P is total profits, and Y is national income. Total savings is composed of savings by workers and by capitalists, where capitalists are a class whose members receive income only from profits:

S = S_w + S_c

S is total savings. S_w is workers' savings, and S_c is capitalist savings. Profits are also split into two parts:

P = P_w + P_c,

where P_w is returns on the capital owned by the workers, and P_c is the return on the capital owned by the capitalists. The behavior assumption is made that both workers and capitalists save a (different) constant proportion of their income:

S_c = s_c P_c

S_w = s_w (W + P_w)

s_c is the capitalists' (marginal and average) propensity to save. s_w is the workers' (marginal and average) propensity to save. The propensities to save are assumed to lie between zero and one and to be in the following order:

0 ≤ s_w < s_c ≤ 1

Workers' savings are assumed to be insufficient to fund all the investment occurring along a steady-state growth path.

The value of the capital stock is divided up into that owned by the workers and by the capitalists:

K = K_w + K_c,

where K is the value of the capital stock, K_w is the value of the capital stock owned by the workers, and K_c is the value of the capital stock owned by the capitalists

2.2 Steady State Equilibrium Conditions

Along a steady-state growth path, in this model, all capital earns the same rate of profits, r:

r = P/K = P_c/K_c = P_w/K_w

It follows from the above set of equations that the ratio of the profits received from the workers to the profits received by the capitalists is equal to the ratio of the value of capital that each class owns:

P_w/P_c = K_w/K_c

Likewise, one can find the ratio of total profits to the profits obtained by the capitalists:

P/P_c = K/K_c

The analysis is restricted to steady-state growth paths where the value of the capitalists' capital and the value of the workers' capital is growing at the same rate:

S/K = S_c/K_c = S_w/K_w

The ratio of profits to savings is the same for the economy as a whole and for workers:

P/S = (P/K)/(S/K) = (P_c/K_c)/(S_c/K_c) = P_c/S_c

Or, after a similar logical deduction for workers:

P/S = P_c/S_c = P_w/S_w

Along a steady-state growth path, planned investment, I equals savings:

I = S

2.3 Deduction of the Cambridge Equation

The following is a series of algebraic substitutions based on the above:

P/I = P/S = P_c/S_c = P_c/(s_c P_c) = 1/s_c

Or:

P = (1/s_c) I

The share of profits in national income is determined by the savings propensity of the capitalists and the ratio of investment to national income:

(P/Y) = (1/s_c) (I/Y)

Recall that the rate of profits is the ratio of profits to the value of capital:

r = P/K = (1/s_c) (I/K)

Recognizing that I/K is the rate of growth, g, one obtains the famous Cambridge equation:

r = g/s_c

As long as the capitalists consume at least some of their income, the rate of profits is greater than the rate of growth along a steady-state growth path. And along such a path the share of income going to profits will be constant.

3.0 Discussion

If one assumes given investment decisions, the Cambridge Equation tells us what rate of profit is compatible with a steady state growth path in which the expectations underlying those investment decisions are satisfied.

Consider two steady states in which the same rate of growth is being obtained. Suppose that along one path workers have a higher propensity to save. Within broad limits, this greater willingness to save among workers has no effect on determining either the share of profits in income or the rate of profits. Only the capitalists' saving propensity matters for the steady state rate of profits, given the rate of growth. Would a capitalist economy have a tendency to approach such a growth path, given a sufficient length of time? I think such stability would entail the evolution of institutions, conventions, the labor force, and what is seen as common sense, including among dominant political parties.

The above model might have some relevance to current political economy discussions elsewhere.

Jean Tirole, A Practitioner Of New Industrial Organization

I have occasionally summarized certain aspects of microeconomics, concentrating on markets that are not perfectly competitive. Further developments along these lines can be found in the theory of Industrial Organization.

One can distinguish in the literature two approaches to IO know as old IO and new IO. Old IO extends back to the late 1950s. Joe Bain and Paolo Sylos Labini laid the foundations to this approach, and they were heralded by Franco Modigliani. I have not read any of Bain and only a bit of Sylos Labini. Sylos was a Sraffian and quite critical of neoclassical economics. He also had interesting things to say about economic development.

As I understand it, new IO consists of applying game theory to imperfectly competitive and oligopolistic markets. I gather new IO took off in the 1980s. Jean Tirole, the winner of this year's "Nobel" prize in economics, is a prominent exponent of new IO.

One can tell interesting stories about corporations with both old IO and new IO. For example, Tirole has had something to say about vertical integration which, based on what I've read in the popular press, might be of interest to me. (Typically, when I explore the theory of vertical integration, following Luigi Pasinetti, the integration is only notional, not at the more concrete level of concern in IO.)

I wonder, though, whether economists can point to empirical demonstrations of the superiority of new IO over old IO. Or have economists studying IO come to embrace new IO more because of the supposed theoretical rigor of game theory? Are specialists in IO willing to embrace the indeterminism that arises in game theory, what with the variety of solution concepts and the existence of multiple equilibria in many games? Or do they insist on closed models with unique equilibria?

References

• Franco Modigliani (1958). New developments on the Oligopoly Front, Journal of Political Economy, V. 66, No. 3: pp. 215-232.

Update (same day): Corrected a glitch in the title. Does this Paul Krugman post read as a direct response to my post?

Noncommunicating Literatures?

During the 20th century, a number of economists more or less independently developed ideas associated with input-output analysis, activity analysis, modeling the economy as a self-sustaining circular flow, and the revival of classical political economy. I think of:

• Leonid Kantorovich: The Soviet economist who shared the 1975 Nobel Memorial Prize in Economic Sciences with Tjalling Koopmans.
• Wassily Leontief: Always emphasized developing an empirically operational version of these ideas.
• Father Maurice Potron: I stumbled across two references to him. I know nothing otherwise about his work.
• Walter Isard: Extended input-output analysis to regional economics.
• Richard Stone: Developed the idea of a Social Accounting Matrix and conventions for national income accounting.
• Jacob Schwartz: Criticized the mainstream economics of his time on the basis of linear economic models.
• Piero Sraffa: Criticized the mainstream economics of his time on the basis of linear economic models.
• John Von Neumann: A mathematician, not an economist.

I wonder how many make connections between the scholarly literature building on the work of each of these researchers. I am not at all sure anybody explicitly and consciously built on Potron or Schwartz.

References

• Wassily W. Leontief (1936). Quantitative Input and Output Relations in the Economic Systems of the United States, Review of Economic Statistics, V. 18, N. 3 (Aug). pp. 105-125.
• Walter Isard (1951) Interregional and Regional Input-Output Analysis: A Model of a Space-Economy, Review of Economics and Statistics, V. 33, No. 4 (Nov.): pp. 318-328.
• Jacob T. Schwartz (1961). Lectures on the Mathematical Method in Analytical Economics, Gordon and Breach.
• Piero Sraffa (1960). Production of Commodities by Means of Commodities: A Prelude to a Critique of Economic Theory, Cambridge University Press.
• J. Ricard N. Stone (1966). The Social Accounts from a Consumer Point of View, Review of Income and Wealth, V. 12, Iss. 1 (Mar.): pp. 1-33. [I HAVEV'T READ THIS OR ANYTHING ELSE BY STONE]
• John von Neumann (1945-1946) A Model of General Economic Equilibrium, Review of Economic Studies, V. 13, No. 1: pp. 1-9.